Special Lagrangian Submanifolds with Isolated Conical Singularities. II. Moduli spaces

Author: Joyce D.

Source: Annals of Global Analysis and Geometry, Volume 25, Number 4, June 2004 , pp. 301-352(52)

Publisher: Springer

Abstract:

This is the second in a series of five papers studying special Lagrangian submanifolds (SLV m-folds) X in (almost) Calabi–Yau m-folds M with singularities X₁, , X_n locally modelled on special Lagrangian cones C₁, , C_n in \mathbb C^M with isolated singularities at 0. Readers are advised to begin with Paper V.

This paper studies the deformation theory of compact SL M-folds X in M with conical singularities. We define the moduli space <IMG SRC="http://images.ingentaselect.com/absimages/klu/0232704x/klu_agag_2004_25_4_5256182h.2.gif" ALT="\mathcal M" TEXT="a mathematical formula">_X of deformations of X in M, and construct a natural topology on it. Then we show that <IMG SRC="http://images.ingentaselect.com/absimages/klu/0232704x/klu_agag_2004_25_4_5256182h.2.gif" ALT="\mathcal M" TEXT="a mathematical formula">_X is locally homeomorphic to the zeroes of a smooth map : <IMG SRC="http://images.ingentaselect.com/absimages/klu/0232704x/klu_agag_2004_25_4_5256182h.3.gif" ALT="\mathcal I" TEXT="a mathematical formula">_X <IMG SRC="http://images.ingentaselect.com/absimages/klu/0232704x/klu_agag_2004_25_4_5256182h.4.gif" ALT="\mathcal O" TEXT="a mathematical formula">_X between finite-dimensional vector spaces.

Here the infinitesimal deformation space <IMG SRC="http://images.ingentaselect.com/absimages/klu/0232704x/klu_agag_2004_25_4_5256182h.3.gif" ALT="\mathcal I" TEXT="a mathematical formula">_X depends only on the topology of X, and the obstruction space <IMG SRC="http://images.ingentaselect.com/absimages/klu/0232704x/klu_agag_2004_25_4_5256182h.4.gif" ALT="\mathcal O" TEXT="a mathematical formula">_X only on the cones C₁, , C_n at X₁, , X_n. If the cones C_i are stable then <IMG SRC="http://images.ingentaselect.com/absimages/klu/0232704x/klu_agag_2004_25_4_5256182h.4.gif" ALT="\mathcal O" TEXT="a mathematical formula">_X is zero, and <IMG SRC="http://images.ingentaselect.com/absimages/klu/0232704x/klu_agag_2004_25_4_5256182h.2.gif" ALT="\mathcal M" TEXT="a mathematical formula">_X is a smooth manifold. We also extend our results to families of almost Calabi–Yau structures on M.

Keywords: Calabi–Yau manifold; special Lagrangian submanifold; singularity

Document Type: Research article

DOI: http://dx.doi.org/10.1023/B:AGAG.0000023230.21785.8d

Affiliations: 1: Lincoln College, T Street, Oxford OX1 3DR, Oxford, U.K., Email: dominic.joyce@lincoln.ox.ac.uk

Publication date: 2004-06-01

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